Methods Summary

In real-time, dividing deaths-to-date by cases-to-date leads to a biased estimate of the case fatality ratio (CFR), because this calculation does not account for delays from confirmation of a case to death, and under-reporting of cases.
Using the distribution of the delay from hospitalisation-to-death for cases that are fatal, we can estimate how many cases so far are expected to have known outcomes (i.e. death or recovery), and hence adjust the naive estimates of CFR to account for these delays.
The adjusted CFR does not account for under-reporting. However, the best available estimates of CFR (adjusting or controlling for under-reporting) are in the 1% - 1.5% range [1–4]. Large studies in China and South Korea estimating the CFR at 1.38% (95% CrI: 1.23–1.53%)[2] and 1.4% (95% CrI: 1.2-1.7%)[4] respectively. Based on these studies, and for simplicity, we assume a baseline CFR of 1.4% for our analysis.
If a country has an adjusted CFR that is higher (e.g. 20%), it suggests that only a fraction of cases have been reported (in this case, 1.420=7.0% cases reported approximately).

Current estimates for percentage of symptomatic cases reported for countries with greater than ten deaths

Temporal variation

Figure 1: Temporal variation in reporting rate. We calculate the percentage of symptomatic cases reported on each day a country has had more than ten deaths. We then fit a Gaussian Process (GP) to these data (see Temporal variation model fitting section for details), highlighting the temporal trend of each countries reporting rate. The red shaded region is the 95% CrI of fitted GP.

Adjusted symptomatic case estimates

Figure 2: Estimated number of new symptomatic cases, calculated using our temporal under-reporting estimates. We adjust the reported case numbers each day - for each country with an under-reporting estimate - using our temporal under-reporting estimates to arrive at an estimate of the true number of symptomatic cases each day. The shaded blue region represents the 95% CrI, calcuated directly using the 95% CrI of the temporal under-reporting estimate.

Reported cases

Figure 3: Reported number of cases each day, pulled from the ECDC and plotted against time for comparison with our estimated true numbers of symptomatic cases each day, adjusted using our under-reporting estimates.

Table of current estimates

Country	Percentage of symptomatic cases reported (95% CI)	Total cases	Total deaths
Afghanistan	38% (27%-53%)	5,226	130
Albania	79% (39%-100%)	880	31
Algeria	35% (23%-50%)	6,253	522
Andorra	23% (13%-39%)	760	49
Argentina	21% (16%-28%)	6,866	329
Armenia	74% (50%-98%)	3,718	48
Australia	86% (52%-100%)	6,975	98
Austria	39% (25%-61%)	15,964	624
Azerbaijan	85% (59%-100%)	2,758	35
Bahamas	48% (11%-99%)	94	11
Bangladesh	96% (75%-100%)	17,822	269
Belarus	99% (93%-100%)	25,825	146
Belgium	12% (9.5%-15%)	53,981	8,843
Bolivia	20% (15%-28%)	3,148	142
Bosnia and Herzegovina	10% (5.9%-16%)	2,181	120
Brazil	13% (10%-15%)	188,974	13,149
Bulgaria	26% (18%-40%)	2,069	96
Burkina Faso	23% (14%-40%)	773	51
Cameroon	74% (22%-100%)	2,800	136
Canada	14% (11%-17%)	72,278	5,304
Chad	5.7% (3.4%-10%)	372	42
Chile	90% (75%-99%)	34,381	346
China	99% (100%-100%)	84,024	4,637
Colombia	26% (20%-33%)	12,930	509
Congo	63% (22%-100%)	341	11
Cote dIvoire	85% (57%-100%)	1,912	24
Croatia	20% (11%-32%)	2,213	94
Cuba	33% (21%-51%)	1,810	79
Cyprus	73% (38%-100%)	905	17
Czechia	35% (26%-46%)	8,269	290
Democratic Republic of the Congo	24% (12%-42%)	1,242	50
Denmark	34% (24%-46%)	10,667	533
Dominican Republic	51% (38%-65%)	11,196	409
Ecuador	5.9% (4.8%-7.2%)	30,486	2,334
Egypt	31% (23%-40%)	10,431	556
El Salvador	51% (26%-91%)	1,112	20
Estonia	34% (21%-52%)	1,751	61
Finland	38% (23%-58%)	6,054	284
France	9.5% (7.7%-11%)	140,734	27,074
Germany	24% (19%-29%)	172,239	7,723
Ghana	95% (81%-100%)	5,408	24
Greece	32% (20%-50%)	2,760	155
Guatemala	44% (25%-73%)	1,342	29
Guernsey	42% (13%-95%)	252	13
Honduras	20% (13%-32%)	2,255	123
Hungary	11% (8%-16%)	3,380	436
Iceland	87% (55%-100%)	1,802	10
India	35% (28%-42%)	78,003	2,549
Indonesia	26% (18%-35%)	15,438	1,028
Iran	34% (28%-41%)	112,725	6,783
Iraq	55% (27%-92%)	3,032	115
Ireland	28% (21%-37%)	23,401	1,497
Isle of Man	21% (7.9%-61%)	332	23
Israel	82% (63%-99%)	16,548	264
Italy	15% (12%-18%)	222,104	31,106
Japan	18% (13%-24%)	16,079	687
Jersey	14% (6.6%-33%)	296	27
Kazakhstan	97% (85%-100%)	5,571	32
Kenya	19% (11%-29%)	737	40
Kosovo	46% (26%-77%)	919	29
Kuwait	84% (61%-100%)	11,028	82
Kyrgyzstan	74% (36%-100%)	1,082	12
Latvia	53% (25%-95%)	951	19
Lebanon	55% (29%-96%)	878	26
Liberia	36% (8.7%-91%)	213	20
Lithuania	40% (24%-65%)	1,505	54
Luxembourg	51% (33%-70%)	3,904	103
Malaysia	95% (76%-100%)	6,779	111
Mali	24% (15%-38%)	758	44
Mauritius	60% (16%-100%)	332	10
Mexico	7.9% (6.5%-9.5%)	40,186	4,220
Moldova	29% (21%-38%)	5,406	199
Morocco	95% (79%-100%)	6,512	188
Netherlands	16% (13%-20%)	43,211	5,562
New Zealand	46% (22%-85%)	1,147	21
Niger	18% (7.9%-32%)	860	49
Nigeria	25% (18%-33%)	4,971	164
North Macedonia	26% (17%-42%)	1,694	95
Norway	38% (16%-67%)	8,158	229
Oman	95% (80%-100%)	4,019	17
Pakistan	43% (33%-54%)	35,788	770
Panama	49% (35%-65%)	8,944	256
Paraguay	84% (49%-100%)	740	11
Peru	42% (33%-51%)	76,306	2,169
Philippines	21% (16%-26%)	11,618	772
Poland	26% (20%-33%)	17,204	861
Portugal	40% (31%-50%)	28,132	1,175
Puerto Rico	33% (21%-49%)	2,329	115
Qatar	93% (68%-100%)	26,539	14
Romania	18% (14%-22%)	16,002	1,016
Russia	90% (78%-99%)	242,271	2,212
San Marino	75% (40%-100%)	643	41
Saudi Arabia	99% (93%-100%)	44,830	273
Serbia	87% (61%-100%)	10,295	222
Singapore	94% (74%-100%)	25,346	21
Sint Maarten	13% (4.2%-34%)	77	15
Slovakia	64% (37%-98%)	1,469	27
Slovenia	17% (11%-25%)	1,463	103
Somalia	26% (15%-46%)	1,219	52
South Africa	45% (34%-59%)	12,074	219
South Korea	42% (17%-79%)	10,991	260
Spain	16% (13%-19%)	228,691	27,104
Sudan	18% (11%-27%)	1,818	90
Sweden	15% (11%-18%)	27,909	3,460
Switzerland	25% (19%-32%)	30,330	1,563
Thailand	74% (50%-99%)	3,017	56
Tunisia	54% (25%-97%)	1,032	45
Turkey	66% (53%-79%)	143,114	3,952
Ukraine	36% (26%-50%)	16,425	439
United Arab Emirates	88% (67%-100%)	20,386	206
United Kingdom	16% (13%-19%)	229,705	33,186
United Republic of Tanzania	43% (22%-79%)	509	21
United States of America	33% (27%-39%)	1,390,746	84,133
Uruguay	42% (20%-83%)	719	19
Uzbekistan	91% (66%-100%)	2,612	11
Venezuela	82% (44%-100%)	440	10

Table 1: Estimates for the proportion of symptomatic cases reported in different countries using cCFR estimates based on case and death timeseries data from the ECDC. Total cases and deaths in each country is also shown. Confidence intervals calculated using an exact binomial test with 95% significance.

Adjusting for outcome delay in CFR estimates

During an outbreak, the naive CFR (nCFR), i.e. the ratio of reported deaths date to reported cases to date, will underestimate the true CFR because the outcome (recovery or death) is not known for all cases [5]. We can therefore estimate the true denominator for the CFR (i.e. the number of cases with known outcomes) by accounting for the delay from confirmation-to-death [1].

We assumed the delay from confirmation-to-death followed the same distribution as estimated hospitalisation-to-death, based on data from the COVID-19 outbreak in Wuhan, China, between the 17th December 2019 and the 22th January 2020, accounting right-censoring in the data as a result of as-yet-unknown disease outcomes (Figure 1, panels A and B in [7]). The distribution used is a Lognormal fit, has a mean delay of 13 days and a standard deviation of 12.7 days [7].

To correct the CFR, we use the case and death incidence data to estimate the proportion of cases with known outcomes [1,6]:

\[ u_{t} = \frac{ \sum_{j = 0}^{t} c_{t-j} f_j}{c_t}, \]

where \(u_t\) represents the underestimation of the proportion of cases with known outcomes [1,5,6] and is used to scale the value of the cumulative number of cases in the denominator in the calculation of the cCFR, \(c_{t}\) is the daily case incidence at time, \(t\) and \(f_t\) is the proportion of cases with delay of \(t\) between confirmation and death.

Approximating the proportion of symptomatic cases reported

At this stage, raw estimates of the CFR of COVID-19 correcting for delay to outcome, but not under-reporting, have been calculated. These estimates range between 1% and 1.5% [1–3]. We assume a CFR of 1.4% (95% CrI: 1.2-1.7%), taken from a recent large study [3], as a baseline CFR. We use it to approximate the potential level of under-reporting in each country. Specifically, we perform the calculation \(\frac{1.4\%}{\text{cCFR}}\) of each country to estimate an approximate fraction of cases reported.

Temporal variation model fitting

We estimate the level of under-reporting on every day for each country that has had more than ten deaths. We then fit a Gaussian Process (GP) model using the library greta and greta.gp. The parameters we fit and their priors are the following: \[ \begin{aligned} &\sigma \sim \text{Log Normal(-1, 1)}: \quad &\text{Variance of the reporting kernel} \\ &\text{L} \sim \text{Log Normal(4, 0.5)}: \quad &\text{Lengthscale of the reporting kernel} \\ &\sigma_{\text{obs}} \sim \text{Truncated Normal(0, 0.5)}, \quad &\text{Variance of the obseration kernel, truncated at 0} \end{aligned} \] The kernel is split into two components: the reporting kernel \(R\), and the observation kernel \(O\). The reporting component has a standard squared-exponential form. For the observation component, we use an i.i.d. noise kernel to acccount for observation overdispersion, which can smooth out overly clumped death time-series. This is important as some countries have been known to report an unusually large number of deaths on a single day, due to past under-reporting.

In the sampling and fitting process, we calculate the expected number of deaths at each time-point, given the baseline CFR. We then use a Poisson likelihood, where the expected number of deaths is the rate of the Poisson likelihood, given the observed number of deaths

Adjusting case counts for under-reporting

We adjust the reported number of cases each day, pulled from the ECDC. Specifically, we divide the case numbers of each day by our “proportion of cases reported” estimates that we calculate each day for each country.*

Limitations

Implicit in assuming that the under-reporting is \(\frac{1.4\%}{\text{cCFR}}\) for a given country is that the deviation away from the assumed 1.4% CFR is entirely down to under-reporting. In reality, burden on healthcare system is a likely contributing factor to higher than 1.4% CFR estimates, along with many other country specific factors.

The following is a list of the other prominent assumptions made in our analysis:

We assume that people get tested upon hospitalisation. A few examples where this is not the case are Germany and South Korea, where people can get tested earlier.
We assume that hospitalisation to death from early Wuhan is representative of the all the other countries (by using the distribution parameterised using early Wuhan data) and that all countries have the same risk and age profile as Wuhan.
Severity of COVID-19 is known to increase with age. Therefore, countries with older populations will naturally see higher death rates. We are extending this analysis to adjust for the age distribution for countries with more than five reported deaths and where age distribution data is available.
All results are linked and biased by the baseline CFR, assumed at 1.4% [3].
The under-reporting estimate is sensitive to the baseline CFR, meaning that small errors in it lead to large errors in the estimate for under-reporting.
There are several sources of uncertainty in this analysis: the reported mean and SD of the delay distribution; the cCFR for each country (on each day) and the baseline CFR. We use the lower and upper mean and SD of the delay distribution to produce the widest interval. The other two uncertainties are just carried through directly in the subsequent calculations, which is crude. An ongoing extension is developing this into a fully Bayesian model, which deals with these different uncertainties more rigorously.

Code and data availability

The code is publically available at https://github.com/thimotei/CFR_calculation. The data required for this analysis is a time-series for both cases and deaths, along with the corresponding delay distribution. We scrape this data from ECDC, using the NCoVUtils package [8].

Acknowledgements

The authors, on behalf of the Centre for the Mathematical Modelling of Infectious Diseases (CMMID) COVID-19 working group, wish to thank DSTL for providing the High Performance Computing facilities and associated expertise that has enabled these models to be prepared, run and processed and in an appropriately-rapid and highly efficient manner.

References

1 Russell TW, Hellewell J, Jarvis CI et al. Estimating the infection and case fatality ratio for covid-19 using age-adjusted data from the outbreak on the diamond princess cruise ship. medRxiv 2020.

2 Verity R, Okell LC, Dorigatti I et al. Estimates of the severity of covid-19 disease. medRxiv 2020.

3 Guan W-j, Ni Z-y, Hu Y et al. Clinical characteristics of coronavirus disease 2019 in china. New England Journal of Medicine 2020.

4 Shim E, Mizumoto K, Choi W et al. Estimating the risk of covid-19 death during the course of the outbreak in korea, february-march, 2020. medRxiv 2020.

5 Kucharski AJ, Edmunds WJ. Case fatality rate for ebola virus disease in west africa. The Lancet 2014;384:1260.

6 Nishiura H, Klinkenberg D, Roberts M et al. Early epidemiological assessment of the virulence of emerging infectious diseases: A case study of an influenza pandemic. PLoS One 2009;4.

7 Linton NM, Kobayashi T, Yang Y et al. Incubation period and other epidemiological characteristics of 2019 novel coronavirus infections with right truncation: A statistical analysis of publicly available case data. Journal of Clinical Medicine 2020;9:538.

8 Abbott S MJ Hellewell J. NCoVUtils: Utility functions for the 2019-ncov outbreak. doi:105281/zenodo3635417 2020.

Aim